Le's conjecture for $x^2-D=4p^n$

Let $D$ be an odd positive integer, let $p$ be an odd prime, and let $N(-D,4,p)$ denote the number of solutions $(x,n)\in\mathbb{N}^2$ of $x^2-D=4p^n$. Let cases $(i)$ and $(ii)$ be the two exceptional cases for $(D,p)$ specified in the source. Le's conjecture. If $D\neq 4p^r+1$, then

$$N(-D,4,p)=3$$

when $(D,p)$ is in case $(i)$ or $(ii)$; otherwise,

$$N(-D,4,p)\leq 2.$$

The survey gives partial bounds but no general proof.